Optimal. Leaf size=494 \[ -\frac {\left (-24 i a^2+36 a+17 i\right ) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{24 b^3}+\frac {\left (-24 i a^2+36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}-\frac {\left (-24 i a^2+36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}+\frac {\left (-24 i a^2+36 a+17 i\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}-\frac {\left (-24 i a^2+36 a+17 i\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}-\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2} \]
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Rubi [A] time = 0.38, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5095, 90, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\left (-24 i a^2+36 a+17 i\right ) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{24 b^3}+\frac {\left (-24 i a^2+36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}-\frac {\left (-24 i a^2+36 a+17 i\right ) \log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{16 \sqrt {2} b^3}+\frac {\left (-24 i a^2+36 a+17 i\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}-\frac {\left (-24 i a^2+36 a+17 i\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{8 \sqrt {2} b^3}+\frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{12 b^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 204
Rule 211
Rule 240
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5095
Rubi steps
\begin {align*} \int e^{\frac {3}{2} i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1+i a+i b x)^{3/4}}{(1-i a-i b x)^{3/4}} \, dx\\ &=\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}+\frac {\int \frac {(1+i a+i b x)^{3/4} \left (-1-a^2-\frac {1}{2} (3 i+8 a) b x\right )}{(1-i a-i b x)^{3/4}} \, dx}{3 b^2}\\ &=-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17-36 i a-24 a^2\right ) \int \frac {(1+i a+i b x)^{3/4}}{(1-i a-i b x)^{3/4}} \, dx}{24 b^2}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17-36 i a-24 a^2\right ) \int \frac {1}{(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx}{16 b^2}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a-i b x}\right )}{4 b^3}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^3}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^3}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 b^3}+\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}+\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}+\frac {\left (17 i+36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}+\frac {\left (17 i+36 a-24 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}\\ &=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}+\frac {\left (17 i+36 a-24 i a^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}+\frac {\left (17 i+36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{16 \sqrt {2} b^3}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 121, normalized size = 0.24 \[ \frac {\sqrt [4]{-i (a+b x+i)} \left (2 i 2^{3/4} \left (24 a^2+36 i a-17\right ) \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};-\frac {1}{2} i (a+b x+i)\right )-i (i a+i b x+1)^{3/4} \left (8 a^2+a (4 b x-5 i)-4 b^2 x^2+7 i b x+3\right )\right )}{12 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 561, normalized size = 1.14 \[ \frac {3 \, b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} \log \left (\frac {b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} + {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} \log \left (-\frac {b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} - {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} \log \left (\frac {b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} + {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) + 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} \log \left (-\frac {b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} - {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (8 i \, b^{2} x^{2} - 2 \, {\left (4 i \, a - 7\right )} b x + 8 i \, a^{2} - 46 \, a - 23 i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{48 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )^{\frac {3}{2}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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